Chapter 12: Q12.50P (page 564)

Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if $t^{μ v}$ is symmetric, show that ${\bar{t}}^{μ v}$ is also symmetric, and likewise for antisymmetric).

Short Answer

Expert verified

The symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation.

Step by step solution

Expression for the property of symmetric and antisymmetric tensor:

Write the property of the symmetric tensor.

$t^{μv} = t^{vμ}$

Write the property of an anti-symmetric tensor.

$t^{μv} = - t^{vμ}$

Here, a negative sign for anti-symmetric tensor.

Determine the symmetry or antisymmetry of a tensor:

Consider $t^{- K λ} = Λ_{μ}^{K} Λ_{v}^{λ} t^{μ v}$

Here, $Λ$ is the Lorentz transformation matrix.

Since $μ$ and v both are summed from $0$ to $3$ , both the values can be interchanged.

Hence, the above equation becomes,

$\begin{matrix} t^{- λ K} = Λ_{μ}^{K} Λ_{μ}^{K} t^{μ v} \\ t^{- λ K} = Λ_{v}^{λ} Λ_{μ}^{K} t^{v μ} \\ t^{- λ K} = Λ_{μ}^{K} Λ_{μ}^{λ} (- t^{μ v}) \\ t^{- λ K} = \pm t^{- K λ} \end{matrix}$

Therefore, the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation.

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Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if $t^{μ v}$ is symmetric, show that ${\bar{t}}^{μ v}$ is also symmetric, and likewise for antisymmetric).

Short Answer

Step by step solution

Expression for the property of symmetric and antisymmetric tensor:

Determine the symmetry or antisymmetry of a tensor:

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Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if tμvis symmetric, show thatt¯μv is also symmetric, and likewise for antisymmetric).

Short Answer

Step by step solution

Expression for the property of symmetric and antisymmetric tensor:

Determine the symmetry or antisymmetry of a tensor:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Translational Dynamics

Astrophysics

Nuclear Physics

Circular Motion and Gravitation

Scientific Method Physics

Study anywhere. Anytime. Across all devices.

Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if $t^{μ v}$ is symmetric, show that ${\bar{t}}^{μ v}$ is also symmetric, and likewise for antisymmetric).